Considerable amount of works has been proposed for estimating \(\Psi \)  [1]}, [2]}. One of the commonly used estimators for \(\Psi \)  is based on the truncated inverse of the empirical covariance operator \(\widehat{\mathcal {C} }_{XX} = \frac{1}{N}\sum _{i=1}^N x_i\otimes x_i\) ; see [3]}, [4]}. Suppose the covariance operator admits the eigenvalue-eigenfunction decomposition \(\widehat{\mathcal {C} }_{XX} = \sum _{r=1}^\infty \hat{\sigma }_r^2\cdot \hat{\theta }_r \otimes \hat{\theta }_r\) . Its rank-\(R\)  truncated inverse is defined as \(\widehat{\mathcal {C} }_{XX}^+ = \sum _{r=1}^R (1/\hat{\sigma }_r^2)\cdot \hat{\theta }_r \otimes \hat{\theta }_r\) . Based on this, the regression function is estimated by \(\widehat{\Psi }= \widehat{\mathcal {C} }_{XX}^{+} \hat{z}\)  with \(\hat{z} = (1/N) \sum _{n=1}^N Y_n x_n\) .
